PUPT 1594, IASSNS 95/111 cond-mat/9602112
arXiv:cond-mat/9602112v3 13 Mar 1996
February 1996
Possible Electronic Structure of Domain Walls in Mott Insulators
Chetan Nayak
⋆
Department of Physics Joseph Henry Laboratories Princeton University Princeton, N.J. 08544
Frank Wilczek
†
School of Natural Sciences Institute for Advanced Study Olden Lane Princeton, N.J. 08540
⋆ Research supported in part by a Fannie and John Hertz foundation fellowship.
[email protected] † Research supported in part by DOE grant DE-FG02-90ER40542.
[email protected]
ABSTRACT We discuss the quantum numbers of domain walls of minimal length induced by doping Mott insulators, carefully distinguishing between holon and hole walls. We define a minimal wall hypothesis that uniquely correlates the observed spatial structure with the doping level for the low-temperature commensurate insulating state of La2−x Bax CuO4 and related materials at x = 18 . We remark that interesting walls can be supported not only by conventional antiferromagnetic but also by orbital antiferromagnetic (staggered flux phase, d-density) bulk order. We speculate on the validity of the minimal wall hypothesis more generally, and argue that it plausibly explains several of the most striking anomalous features of the cuprate high-temperature superconductors.
2
It is familiar in several contexts that spatial defects, and specifically domain walls, can have a profound effect upon electronic behavior. This has perhaps been most thoroughly documented in the case of polyacetylene [1, 2], where (as we shall momentarily recall) it leads directly to exotic quantum numbers for the elementary excitations. Shortly after polyacetylene was analyzed, there was a resurgence of interest in the classic Mott insulator problem – spurred by the discovery of the high-Tc copper-oxide superconductors – with particular emphasis on the nature of the low-lying excitations which result upon doping. Several proposals revolve around the idea that these excitations are solitonic. Recent experiments on La2−x Bax CuO4 and related materials exhibit striking commensurability effects which, we shall argue, acquire a simple but profound significance within this circle of ideas. Our analysis encourages us to consider generalizing these ideas so as to address the problem of anomalous behaviors in the copper oxides more generally, when commensurability fails. On a qualitative level, at least, the results appear quite encouraging.
1. Quantum Numbers of Minimal Walls Let us briefly recall the main results on quantum numbers of domain wall defects in polyacetylene and related one dimensional substances where a Z2 symmetry breaking opens a gap at half filling. They can be modelled adequately, as regards quantum numbers, using a simple scalar field φ to represent the order pa¯ rameter, and its interactions with the electrons by the Yukawa term g ψψφ. The order parameter is supposed to take values hφi = ±v in the uniform ground state,
which induces a mass term (gap) for the electrons. If the mass is positive for one sign then formally it is negative for the other; however viewed in itself this sign is meaningless because it can be reabsorbed into the definition of (one chirality of) ψ. However the change in sign at a domain wall is significant, and leads to the existence of a normalizable zero mode for ψ localized at the wall [3]. This midgap state acquires half its spectral weight from states in the original Dirac sea, and 3
half from above. Thus if it is left unoccupied the state has electron number −1/2
while if it is occupied the state has electron number +1/2.
Inclusion of the electron spin variable changes the picture. There are now two zero-modes, one for each spin. If both are unoccupied, the state is a spin singlet with electron number (and hence charge) −1; if both are occupied the electron
number is +1, and of course the electron number vanishes if one is occupied and the other empty. The charge ±1 states are spin singlets – they represent the full
sea or its complement – while the charge 0 states form a spin doublet. Note that
to reproduce the quantum numbers appropriate to removing an electron – a doped hole – we require two domain walls, and must choose to take the charge −1 ‘holon’
state on one and the charge 0 doublet ‘spinon’ one the other, i.e. occupying exactly one of the four available zero-modes.
This analysis, which can be made completely formal and rigorous, permits a simple pictorial representation (Figure 1). It can also be summarized in the rubric { hole
→
holon + spinon }, with the understanding that both holon and
spinon are associated with wall defects.
With this discussion in mind, it is easy to appreciate an important distinction that arises for walls in two dimensions (Figure 2). For purposes of determining quantum numbers one can consider a vertical two-dimensional wall as a collection of one-dimensional walls – one on each horizontal line – and reduce to the preceding case. In this way we see that the vacant wall, considered in depth by Schulz [4], does not correspond directly to the quantum numbers one would associate with an ordinary hole; rather it gives an array of holons. To represent the degrees of freedom of weakly interacting holes faithfully, one must instead allow the more complicated configuration shown on the right. Note that this configuration corresponds to filling along the wall. Indeed, this is basically the same factor
1 4
1 4
we encountered
two paragraphs ago. Note that there is nothing obviously special about this filling, viewed in itself; its distinction resides only in its pedigree. The pictorial representation in Figure 2 makes the conclusion of the formal 4
analysis quite appealing intuitively. The idealized hole wall schematically indicated there contains equal numbers of two kinds of sites, empty and singly occupied. The empty sites carry charge (relative to the uniform bulk state) but no spin: they carry holon quantum numbers. The singly occupied sites have a spin degree of freedom each of whose orientations is equally favorable, due to an evident symmetry, but no deviation from the uniform bulk density. They carry spinon quantum numbers. Together, each such pair has the quantum numbers of a single hole. When one introduces interactions and hopping to this static picture the low-energy eigenstates will be quite different in structure, but one would not expect the quantum numbers to be altered. It is remarkable that the quantum numbers of hole (not holon) walls are just what is needed to reproduce the spatial structure revealed in neutron scattering by Tranquada et al. for the low-temperature insulating state of La1.6−x Nd0.4 Srx CuO4 at the appropriate commensurate doping x =
1 8
[5]. The neutron scattering results
indicate stripes modulating the bulk spatial structure along every fourth row of atoms, and this will occur precisely at x =
1 8
when the walls are, in the sense we
have defined above, hole domain walls. One could of course imagine adding extra domain wall bits not associated with doping, or conversely hole excitations which are not attached to domain walls. However the narrowness of the observed non-superconducting phase near x =
1 8
implies and is implied by the minimal wall hypothesis: that exactly enough domain wall is produced, at least at low temperatures, to provide a midgap home for the dopants. To avoid possible misunderstanding, let us emphasize that we do not propose it as a general law that the hole quantum numbers must be represented faithfully. One can certainly imagine that the spins of the dopants are frozen, as in the underlying bulk state, or that the charges are frozen. That is, there could be phases where the quantum numbers of either holon or spinon domain walls alone are realized – indeed, it appears [5] that in La2 NiO2 the former possibility occurs. 5
Rather we are proposing, much in the spirit of Landau’s Fermi liquid theory, that there is a universality class corresponding to faithful representation of the weaklycoupled dopant degrees of freedom, and that the minimal realization of this class occurs in the situation mentioned. Presumably the hole domain wall is favored when Coulomb repulsion dominates the antiferromagnetic correlation energy and the holon wall is favored in the opposite case. Our picture of the
1 8
state has significant dynamical consequences. According
to it, the charge carriers in this state are contained in an infinite array of (coupled) 1-dimensional metals. These degrees of freedom should dominate the specific heat and this should be proportional to T . Most interesting and characteristic, however, is the resistivity. Impurities in one-dimensional metals (Luttinger liquids) with repulsive interactions lead to a (non-universal) power-law divergence of the resistivity [6, 7]. By contrast, weak Anderson localization in two dimensions leads to a logarithmic divergence. In the La1.6−x Nd0.4 Srx CuO4 resistivity data of Nakamura and Uchida [8], one sees a slight positive curvature in the in-plane resistivity at temperatures above the structural phase transition which presumably pins the domain walls. In La2−x Srx CuO4 , this positive curvature is the precursor of the logarithmic divergence of weak localization. Below the structural transition, however, there is a striking increase in the divergence of the resistivity. It is attractive to interpret this marked increase as resulting from the effects of impurities on the essentially one-dimensional transport which takes place in pinned domain walls. Some other ideas relating to the existence and significance of domain walls and antiferromagnetic stripes may be found in [9-12].
6
2. Analysis of Single Walls Let us very briefly indicate the shape a more mathematical treatment of the preceding would take. The problem of domain walls in a doped antiferromagnet was discussed by Schulz [4]; the reader should compare [13] where a formally complete but highly compressed discussion of a closely related problem is given. Consider the Hubbard model on a two-dimensional square lattice: H = −t
X
c†α,i cαi + U
X
n↑i n↓i
(2.1)
i
n.n.hiji
where α is a spin index. In the spin-density wave formalism, we assume an antiferromagnetic background with Sz = (−1)m+n S (i = (m, n)). A domain wall solution has Sz = (−1)m+n S for n ≥ 1 and Sz = (−1)(m+n+1) S for n ≤ −1. In the presence of this mean field, the factorized form of the Hamiltonian (2.1) is: H = −t
X
c†α,i cαi + US
X i
n.n.hiji
σ(n)(−1)i (n↑i − n↓i )
(2.2)
where σ(n) = +1 for n ≥ 1, σ(n) = −1 for n ≤ −1, and σ(0) = 0. This single-
particle Hamiltonian may be diagonalized and, as usual, one finds conduction and valence bands split by a gap; the self-consistency condition that the ground state of the Hamiltonian (2.2) exhibit the required antiferromagnetic background leads to a BCS-like gap equation. For our purpose we want to focus on the essentially new states which arise in the presence of the domain wall. Physically, it is not hard to see why such states should exist: an electron located at the wall essentially sits in a potential well in the direction perpendicular to the wall but is free to move along the wall. At the level of the single-particle Hamiltonian (2.2), these states are of the form: ψk (m, n) = eikma ei(k+π)na + i(−1)m+n ei(k+π)na e−κk |na| 7
(2.3)
with sinh(κk a) =
US 2t sin ka
(2.4)
and energy E = 2t cos pa(cosh κk a − 1)
(2.5)
In the small-U limit, the domain-wall band becomes narrow, as was found by Schulz [4]. A more sophisticated treatment must derive the form of the factorized Hamiltonian self-consistently. Such formal arguments can be made to seem more powerful than they are, since they are intrinsically weak-coupling arguments and involve extrapolating to reach any realistic (or even finite) coupling. In calculating digital quantities such as the number of states with specified quantum numbers extrapolation of this sort has a fair chance of success, but even here real physical questions may arise. For example, when we consider oblique hole domain walls two structures naturally suggest themselves, as depicted in Figure 3. The density of holes per unit length of domain wall is twice as great for one as for the other; one or the other, even a superposition might be most favorable, depending on detailed energetics.
3. Alternative Bulk Orders In the preceding section we have framed the discussion in the context of antiferromagnetic bulk ordering. It is important to emphasize, however, that the underlying principles leading to the quantum number and commensurability assignments are more general. In particular, they apply to a different kind of order known in its two-dimensional incarnation variously as orbital antiferromagnetism, staggered flux, or d-density [14,15]. We will use the last name, since the others are inappropriate for our extrapolations outside two dimensions. The issue of other types of ordering is not merely academic. The neutron scattering data of [5] indicate that there is first a transition 8
to a striped phase without spin ordering – perhaps consisting of domain walls in the alternate ordering that we consider below – and then a transition to a phase with antiferromagnetic stripes. For later purposes it is important to define d-density order in a form that applies to other dimensionalities (especially 1!), as follows. Let the lattice spacing be a in all directions. Then we suppose the existence of a non-trivial correlation hc†α (k)cβ (k + Q)i = i δαβ f (k)
(3.1)
where Q has components of magnitude π/a in all directions and f is a real function that changes sign upon π/2 rotation around any axis. This correlation breaks symmetry under time reversal T , translation by one lattice spacing, and π/2 rotations, but in such a way that the square of any of these operations, or the product of any two, is a valid symmetry. This order, although its direct manifestations in neutron scattering (for example) are quite subtle, effectively halves the size of the Brillouin zone and naturally induces Mott insulator behavior at half filling. In two dimensions the energetics for this kind of ordering appear quite favorable, especially in view of the nesting property. Moreover since the symmetry breaking is Z2 one very naturally has topologically stable domain walls in this case, and they support zero modes as analyzed above. At and near half-filling, such a state may be thought of as a d-wave spin-singlet particle-hole paired state, just as a charge-density-wave is an s-wave particle-hole paired state. The i on the right-hand-side of (3.1) is necessitated by the d-wave symmetry of f (k). A BCS-like paired wavefunction may be written down for a state with such ordering: Ψ =
Y k
(uk↑ c†k↑ + vk↑ c†k+Q↑ ) (uk↓ c†k↓ + vk↓ c†k+Q↓ ) |0 > ,
(3.2)
where the product runs over some subset of the interior of the magnetic zone (i.e. the diamond (kx ± ky )a = ±π), and |u|2 + |v|2 = 1 for all values of the indices, to 9
preserve normalization, and the number of factors is determined by the density of electrons (see below). The product uk vk is imaginary; its magnitude is proportional to the gap. In two dimensions, d-density state exhibits staggered flux. In the case of the ansatz wavefunction (3.2), h c†α x+a cxα i =
X k
X vkα ukα e−ik cot a u¯kα vkα −¯ vkα vkα + e−iQ·x e−ik·a u¯kα ukα −¯ k
(3.3)
If the uk ’s and vk ’s are time-reversal invariant, uk = u−k , vk = v−k , then the first sum is purely real while the second is purely imaginary for d-density ordering. The flux through a plaquette with its upper left corner at x is the fourth power of (3.3); the flux through a neighboring plaquette is the complex conjugate. Since the d-density state exhibits staggered flux, it is presumably in the same universality class as the staggered flux state. However, the variational state (3.2) differs from the staggered flux variational states in that the real part of the hopping matrix element is kept fixed and the imaginary part varied in the former case while the magnitude is kept fixed and the phase varied in the latter. Because the ordering in question breaks only a discrete symmetry, it becomes particularly favorable – as compared to potential rival orders breaking a continuous symmetry – in low dimensions, due to its relative immunity from fluctuations. A mean-field analysis of extended Hubbard models with nearest-neighbor repulsion in 2+1 dimensions indicates the favorability of this ordering for a range of parameters. But perhaps the most profound and compelling argument for its physical significance comes from analysis of the 1+1 dimensional Hubbard model near half filling, as we now sketch. The Fermi surface splits into two points, describing left- and right-movers. Spin and charge excitations travel at different velocities, and are conveniently represented by different fields. Let the charge field for left and right movers be represented by the appropriate components χL,R of a scalar field χ compactified on a √ circle of radius 2. At generic values of the filling, these are free fields. Precisely 10
at half filling there is an additional relevant interaction, the Umklapp process, that in terms of the electron fields is Lumklapp = u(ǫαβ ψL†α ψL†β )(ǫγδ ψRγ ψRδ )
(3.4)
and for the holon fields induces the simple form L =
√ 1 (∂χ)2 + 2u cos 2χ . 2
(3.5)
U 16t ,
where t is the hopping parameter. For an attractive √ interaction u < 0 the energy is minimized at hχi = 0 or 2π, and for a repulsive √ interaction u > 0 the energy is minimized at hχi = ±π/ 2. Here χ ≡ χL +χR and u =
√
In terms of the holon variables ψL ≡ e−iχ/
2,
√
ψR ≡ eiχR /
2,
hχi = 0 cor-
† responds to hψR ψL† i = hψL ψR i = c , a real number. This is a conventional, √ commensurate charge density wave. On the other hand hχi = π/ 2 corresponds
to
† i = if , hψR ψL† i = −hψL ψR
(3.6)
a pure imaginary number. As a result of the minus sign this state does not have charge-density order, contributions from L-R and R-L terms cancelling. Instead, (3.6) is precisely of the d-density form. The one-dimensional model suggests another interesting although more speculative possibility. If we assume that spin-charge separation occurs in two dimensions as it does in one, then we can imagine that the holon degrees of freedom undergo d-density ordering, opening a gap for charged excitations but leaving gapless spin excitations. In any case, in the one dimensional model the charged excitations are associated with domain walls. Indeed one has the relationship j0 =
1√ ∂ χ, 2π 2 1
indicating a
unique connection between domain walls and localized charge 1/2 inhomogeneities. 11
Since d-density ordering breaks a Z2 symmetry, domain walls are topologically stable. As we will see below, we can use more powerful arguments than those of section 2 to show that these walls support gapless modes. In order to see that they do, we can calculate the currents that are built up by an external electromagnetic field in the presence of a domain wall, using the method of Goldstone and Wilczek [16]. Equivalently, we can calculate the terms which are generated in the low-energy effective action for the electromagnetic field when the gapped fermionic excitations are integrated out. This action will fail to be gauge-invariant, necessitating the existence of gapless excitations at the domain walls which restore gauge invariance as in the case of the edge excitations in the quantum Hall effect [17]. The result of such a calculation is the effective action:
Z
SstaggeredC.−S. =
3
d k ǫµνλ Fνλ (Q − k)
(cos kx a + cos ky a − 2) Aµ (k) (3.7) + iQµ (cos kx a − 1)Ax (k) + (cos ky a − 1)Ay (k) .
or, in real space:
Sstaggered C.−S. =
Z
3
iQ·x
d x ǫµνλ e
x+aˆ Z µ ηµ Aµ (x + aˆ µ) − Aµ (x) + iQµ A Fνλ (x) x
(3.8)
To see that there must be gapless excitations at a domain wall, suppose to the contrary that there are no low-energy excitations other than photons. This is inconsistent because the action (3.8) is not gauge invariant. Under a gauge transformation, Aµ → Aµ + ∂µ f , the action transforms as: δS =
Z
d x ǫµνλ e
=
Z
d3 x ǫµνλ ∂µ
3
iQ·x
µ) − ∂µ f (x) + iQµ f (x + aˆ µ) − f (x) Fνλ (x) ηµ ∂µ f (x + aˆ ! µ) − f (x) Fνλ (x) . ηµ eiQ·x f (x + aˆ (3.9)
12
Thus, the action fails to be gauge-invariant at the boundary of the d-density ordered region, i.e. at a domain wall. This is rectified, however, if there are gapless modes at the domain wall with the action Sdomain wall =
Z
µ)−ϕ(x) ǫµν Fµν (x) (3.10) d2 x (Dµ ϕ)2 + n ˆ µ ηµ eiQ·x ϕ(x+aˆ
where Dµ is the ordinary covariant derivative, and ϕ transforms as ϕ → ϕ+f Since µ) − Fµν (x) ǫµν ϕ(x), it the latter term can be rewritten as n ˆ µ ηµ eiQ·x Fµν (x + aˆ is clear that this theory describes an ordinary, non-chiral scalar field so long as the electromagnetic fields are slowly varying on distances of order the lattice spacing.
4. Possible Dynamical Implications Given our interpretation of the state at x = 81 , it is natural to speculate that the minimal domain wall hypothesis is valid more generally – that the physical effect unique to this doping lies not in the nucleation of hole walls to accommodate holes, but only in that the resulting walls get locked into a rigid spatial structure. Some recent photoemission data [18] on Bi2 Sr2 CaCu2 O8+δ is most suggestive in this regard [19]. The data suggests the existence of significant flat regions of the π π Fermi surface at or near kx = ± 4a or ky = ± 4a . But these are precisely what one
expects from horizontal and vertical hole walls! Indeed, for a horizontal wall ky is indefinite while for the
1 4
filling characteristic of hole walls the Fermi points occur at
π kx = ± 4a . Several potential complications should be noted. The flat regions occur
do not extend to the smallest values of ky , but that is where one is sampling over
a large geometric region, and the possibility for interwall interactions exists. Also, there could well be low-energy electronic excitations in the bulk state, as occurs for special points on the Fermi surface near half-filling in the d-density and flux π π , ± 2a ). Also, at large k the form-factor of the zero modes will cut phases at (± 2a
down their response from that expected for point-like particles. The appearance 13
and strength of the indicated flat regions, then, requires (on our interpretation) specific properties of the horizontal and vertical walls – in particular, if the flat regions extend to large momenta, one requires that these walls are effectively quite narrow in the transverse direction. The dynamics of interacting domain walls with non-trivial electronic structure appears to be a new and challenging problem in the context of condensed matter physics; it includes a form of finite-density string theory. The discussion that follows represents what we think are plausible conjectures, but is very far from rigorous. An first qualitative consequence of these ideas is that the behavior of the electronic degrees of freedom, being described by a 1+1 dimensional theory, are much more affected by interactions, even at low temperature, than in a homogeneous phase. One must start their description with the Luttinger liquid theory, including spin-charge separation, rather than the Fermi liquid theory. Anderson has championed this point of departure for some time, at least partly on phenomenological grounds [20]. A novel source of electrical resistance in domain wall transport is the interaction of gapless charged excitations which carry currents with transverse vibrations of the domain wall. The natural effective theory for the fermionic excitations interacting with these long-wavelength ‘phonon’ excitations of the wall is: L = Lfermion + ∂t φ
2
− c2 ∂x φ
2
+ λ ψ † ψ ∂x φ
(4.1)
where φ is the domain-wall ‘phonon’ displacement field. We can diagonalize the bosonized form of this theory. It is straightforward to show that the conductivity yielded by such a calculation is of the same form as for a non-interacting electron system, σ(ω) ∝ δ(ω). This runs counter to one’s intuitive expectation that currents
can be dissipated when charge carriers emit phonons. The problem, of course, is that unless the ‘phonons’ interact with other degrees of freedom they return all of their energy and momentum to the charged currents. In our case we expect the 14
phonons to be quite strongly coupled to the bulk degrees of freedom, and to lose their energy before they can return it to the electrons, so σ∼
1 . T
(4.2)
As we have recently emphasized, this is the naive scaling for temperature dependence of conductivity in a theory with a Fermi surface [21]. In a conventional Fermi liquid it is usually modified because the resistance is due to so-called dangerous irrelevant operators, which introduce additional powers of T . In (4.1), all interactions are marginal, so we expect the naive scaling (4.2) to hold. Another issue of great importance is the sensitivity of wall conductivity to impurities. The leading interaction introduced by an impurity is of the type ∆L = δ 1 (x)ψ † ψ ,
(4.3)
and is marginal by power counting. For repulsive electron-electron interactions this operator becomes relevant, and one expects large localization effects, as we have already mentioned in the context of the x =
1 8
state. However when the domain
walls are not pinned, exchange of the wall vibration ‘phonons’, as well as of ordinary phonons, generates a competing attractive interaction. If the attractive interaction wins out, then the impurity interaction is irrelevant [6]. For a random distribution of impurities, the same is true if the attractive interaction is sufficiently strong [7]. Attraction in the Cooper channel, which in higher dimensions triggers an instability toward superconductivity, cannot do so in 1 dimension because of large fluctuation effects. It leads instead to unusually good (but not quite super) conductivity, insensitive to impurities. The divergent resistivity seen in the experiments referred to earlier is presumably the result of the fact that the pinning of the domain walls by a structural phase transition pushes the ‘phonon’ excitations to high energies. As a result, the attractive interaction between electrons is reduced and the repulsive case – with markedly increased sensitivity to impurities – applies. 15
Suppose, now, that there are a number of domain walls in the system. If these walls are close enough charge can freely tunnel from one to another. This is a relevant perturbation. Clarke, Strong, and Anderson [22] have argued that tunneling between one-dimensional Luttinger liquids may be an incoherent process. Then the tunneling conductivity vanishes as T → 0. Even if their arguments were
incorrect, the tunneling between domain walls will be incoherent if the walls are not parallel or are fluctuating since, in such cases, the momentum parallel to the walls cannot be conserved. There are then two possibilities for the resistance between two points in a sample. If the two points are continuously connected by domain walls, then the resistance is simply linear in T . If the two points are not continuously connected by domain walls, then some inter-wall tunneling is necessary for charge to be transported from one point to the other. In such a case, the resistance is linear in T at high temperatures, where the inter-wall tunneling varies slowly with the temperature and is not yet the limiting factor. At low temperatures, the resistance should turn up and diverge as T → 0. As the doping is increased from zero, we expect domain walls to proliferate until, finally, a percolation critical point is reached. If the doping is less than this critical value, then we expect the sample to be metallic at high temperatures with ρ ∼ T as discussed above and then to cross over to insulating behavior at low temperatures. At the percolation threshold, we expect metallic behavior, with
ρ = aT extrapolating to ρ = 0 at T = 0 due to the insensitivity to impurities noted above. The underlying bulk order is destroyed at some doping level at or above the percolation critical point. When this occurs, Fermi liquid behavior is expected. This state of affairs may be summarized by saying that exotic metallic behavior is expected in the “quantum critical regime” [23] centered at the percolation critical point. At any doping level less than that which restores Fermi liquid behavior, c-axis transport, i.e. transport out of the plane, should be qualitatively similar to in-plane transport in the underdoped regime. At high temperatures, ρ ∼ T is expected,
but at low temperatures, the incoherence of tunneling between domain walls in 16
neighboring planes causes insulating behavior. All this is strikingly reminiscent of behavior recently observed experimentally in [24], if we suppose the percolation threshold is close to the optimal doping level for superconductivity. The correlation of maximal Tc with a bifurcation in the qualitative behavior of the low-temperature normal state resistivity is a striking ‘coincidence’, which as far as we know has not previously been illuminated theoretically. It can be motivated, within the circle of ideas we have been advocating, by a plausible extension of the interlayer tunneling hypothesis [25], which has had considerable semiquantitative success, to intralayer, interwall tunneling. Indeed, supposing that the transition to superconductivity is triggered even at the planar level by the gain in delocalization energy when pairs can propagate coherently over the plane, it seems eminently reasonable that the largest gain in accessible area per unit strength of pairing takes place near the percolation threshold, where the accessibility of large areas hangs in delicate balance.
Acknowledgment: We wish to thank N. Bonesteel and J. Axe for calling our attention to the
1 8
effect and for helpful information regarding it; A. Nersesayan
and L. Gorkov for alerting us to the literature on orbital antiferromagnetism [14]; A. H. Castro Neto for calling our attention to other relevant work [9-11]; and G. Baskaran, J. R. Schrieffer and S. Treiman for several instructive discussions.
17
REFERENCES 1. W.P. Su, J.R. Schrieffer, and A.J. Heeger, Phys. Rev. Lett. 42 1698 (1979). 2. A. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su Rev. Mod. Phys. 60, 781 (1988). 3. R. Jackiw, C. Rebbi Phys. Rev. D13, 3398 (1976). 4. H.J. Schulz, J. Physique 50 2833 (1989). 5. J.M. Tranquada et al., Nature 375 561 (1995). 6. C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68 1220 (1992); Phys. Rev. B 46 15233 (1992). 7. T. Giamarchi and H.J. Schulz, Phys. Rev. B 37, 325 (1988). 8. Y. Nakamura and S. Uchida, Phys. Rev. B 46 5841 (1992). 9. J. Zaanen and O. Gunnarson, Phys. Rev. B 40 7391 (1989). 10. D. Poilblanc and T.M. Rice, Phys. Rev. B 39 9749 (1989). 11. V.J. Emery and S.A. Kivelson, Physica C 209 597 (1993); Physica C 235240 189 (1994); in Strongly Correlated Electronic Materials: The Los Alamos Symposium 1993, eds. K.S. Bedell et al., 619, Addison-Wesley, Reading, MA 1994 12. A.H. Castro-Neto and D.W. Hone, Phys. Rev. Lett. in press, condmat/9511079. 13. T. L. Ho, J. Fulco, J. R. Schrieffer, and F. Wilczek Phys. Rev. Lett. 52, 1524 (1984). 14. H.J. Schulz, Phys. Rev. B 39 2940 (1989); A.A. Nersesyan and A. Luther, unpublished; A.A. Nersesyan, G.I. Japaridze, and I.G. Kimeridze, J. Phys. Cond. Mat. 3 3353 (1991); A.A. Nersesyan and G.E. Vachnadze, J. Low Temp. Phys. 77 293 (1989); L. Gorkov and A. Sokol, Phys. Rev. Lett. 69 18
2586 (1992); For a pedagogical review, see “Spin-Singlet Ordering Suggested by Repulsive Interactions,” C. Nayak and F. Wilczek, cond-mat/9510132 15. I. Affleck and B. Marston, Phys. Rev. B 37, 3774 (1988). 16. J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47 986 (1981). 17. X.G. Wen, Phys. Rev. B 43 11025 (1991); Phys. Rev. Lett. 64 2206 (1990). 18. D.S. Marshall, et al., “Unconventional Electronic Structure Evolution with Hole Doping in Bi2 Sr2 CaCu2 O8+δ Angle-resolved Photoemission Data”, Stanford preprint. 19. G. Baskaran, private communication. 20. P. W. Anderson, G. Baskaran, Z. Zhou, and T. Hsu, Phys. Rev. Lett. 58 279 (1987). 21. C. Nayak and F. Wilczek, “ Physical Properties of Metals from a Renormalization Group Standpoint,” Intl. Jour. Mod. Phys. B. (in press), condmat/9507040 22. D.G. Clarke, S.P. Strong, and P.W. Anderson, Phys. Rev. Lett. 72 3218 (1994). 23. S. Sachdev, “Quantum phase transitions in spins systems and the high temperature limit of continuum quantum field theories,” Proceedings of STATPHYS 19, cond-mat/9508080, and references therein. 24. Y. Ando and G.S. Boebinger, AT&T Bell Laboratories Preprint 25. J.M Wheatley, T.C. Hsu, P.W. Anderson, Phys. Rev. B 37 5897 (1988).
19
FIGURE CAPTIONS 1) Holon and spinon walls for underlying antiferromagnetic order in one dimension. Note that in each case there is a mismatch of the ordering at the two ends, compared to the uniform ground state. Simply removing the possibility of occupancy at one site does not faithfully represent the two degrees of freedom associated with an electron. The spinon configuration, which has two degenerate states but zero charge, restores these. 2) Holon versus hole walls for underlying antiferromagnetic order in two dimensions. The double-headed arrows represent the possibility of choosing either spin direction. 3) Alternative simple structures for oblique hole walls.
20
Holon (q=1,s=0)
Figure 1.
Spinon (q=0,s=1/2)
21
22
Holon domain wall
Figure 2.
Hole Domain Wall
23
Figure 3.
